Distributed Lower Bounds for Ruling Sets

Given a graph $G = (V,E)$, an $(\alpha, \beta)$-ruling set is a subset $S \subseteq V$ such that the distance between any two vertices in $S$ is at least $\alpha$, and the distance between any vertex in $V$ and the closest vertex in $S$ is at most $\beta$. We present lower bounds for distributedly computing ruling sets. The results carry over to one of the most fundamental symmetry breaking problems, maximal independent set (MIS), as MIS is the same as a $(2,1)$-ruling set. More precisely, for the problem of computing a $(2, \beta)$-ruling set (and hence also any $(\alpha, \beta)$-ruling set with $\alpha > 2$) in the LOCAL model of distributed computing, we show the following, where $n$ denotes the number of vertices and $\Delta$ the maximum degree. $\bullet$ There is no deterministic algorithm running in $o\left( \frac{\log \Delta}{\beta \log \log \Delta}\right) + o\left(\sqrt{\frac{\log n}{\beta \log \log n}}\right)$ rounds, for any $\beta \in o\left(\sqrt{\frac{\log \Delta}{\log \log \Delta}}\right) + o\left(\left(\frac{\log n}{\log \log n}\right)^{1/3}\right)$. $\bullet$ There is no randomized algorithm running in $o\left( \frac{\log \Delta}{\beta \log \log \Delta}\right) + o\left(\sqrt{\frac{\log \log n}{\beta \log \log \log n}}\right)$ rounds, for any $\beta \in o\left(\sqrt{\frac{\log \Delta}{\log \log \Delta}}\right) + o\left(\left(\frac{\log \log n}{\log \log \log n}\right)^{1/3}\right)$. For $\beta > 1$, this improves on the previously best lower bound of $\Omega(\log^* n)$ rounds that follows from the old bounds of Linial [FOCS'87] and Naor [J.Disc.Math.'91] (resp.\ $\Omega(1)$ rounds if $\beta \in \omega(\log^* n)$). For $\beta = 1$, i.e., for MIS, our results improve on the previously best lower bound of $\Omega(\log^* n)$ \emph{on trees}, as our bounds already hold on trees.